Nuprl Lemma : cubical-type-ap-morph-comp
∀[X:CubicalSet]. ∀[A:{X ⊢ _}]. ∀[I,J,K:Cname List]. ∀[f:name-morph(I;J)]. ∀[g:name-morph(J;K)]. ∀[a:X(I)]. ∀[u:A(a)].
(((u a f) f(a) g) = (u a (f o g)) ∈ A((f o g)(a)))
Proof
Definitions occuring in Statement :
cubical-type-ap-morph: (u a f),
cubical-type-at: A(a),
cubical-type: {X ⊢ _},
cube-set-restriction: f(s),
I-cube: X(I),
cubical-set: CubicalSet,
name-comp: (f o g),
name-morph: name-morph(I;J),
coordinate_name: Cname,
list: T List,
uall: ∀[x:A]. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
cubical-type: {X ⊢ _},
cubical-type-ap-morph: (u a f),
cubical-type-at: A(a),
pi1: fst(t),
pi2: snd(t),
and: P ∧ Q,
squash: ↓T,
all: ∀x:A. B[x],
true: True,
subtype_rel: A ⊆r B,
prop: ℙ,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
equal_wf,
list_wf,
coordinate_name_wf,
cube-set-restriction_wf,
name-comp_wf,
iff_weakening_equal,
cubical-type-at_wf,
name-morph_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
setElimination,
thin,
rename,
productElimination,
sqequalRule,
applyEquality,
lambdaEquality,
imageElimination,
extract_by_obid,
isectElimination,
because_Cache,
hypothesis,
functionExtensionality,
hypothesisEquality,
dependent_functionElimination,
equalitySymmetry,
natural_numberEquality,
imageMemberEquality,
baseClosed,
universeEquality,
equalityTransitivity,
independent_isectElimination,
independent_functionElimination,
isect_memberEquality,
axiomEquality
Latex:
\mforall{}[X:CubicalSet]. \mforall{}[A:\{X \mvdash{} \_\}]. \mforall{}[I,J,K:Cname List]. \mforall{}[f:name-morph(I;J)]. \mforall{}[g:name-morph(J;K)].
\mforall{}[a:X(I)]. \mforall{}[u:A(a)].
(((u a f) f(a) g) = (u a (f o g)))
Date html generated:
2017_10_05-AM-10_12_29
Last ObjectModification:
2017_07_28-AM-11_18_20
Theory : cubical!sets
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